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Full text of "Treatise On Analysis Vol-Ii"

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1 NORMED ALGEBRAS 307 Examples of Narmed Algebras (15.1.4) For each nonempty set X, the set ^C(X) of bounded complex-valued functions on X is a commutative Banach algebra with respect to the norm (7.1.1) and ordinary multiplication of functions: the inequality (15.1.1) is clearly satisfied, the constant function 1 is the unit element of ^CPO and satisfies (15.1.2). If X is a topological space the subspace ^c(X) of bounded continuous functions on X is a closed subalgebra of ^C(X). If X is metrizable, separable and locally compact, the space ^(X) (13.20.5) of continuous func- tions which tend to 0 at infinity, and the space Jf* C(X) of continuous functions with compact support, are ideals in the algebra ^(X), the former being the closure of the latter. (15.1.5) If X is the closed disk \z\ g 1 in C, then the set j/(X) c <jpc(X) of continuous functions on X which are analytic in the interior \z\ < 1 of X is a closed subalgebra of #C(X), by virtue of the theorem of convergence of analytic functions (9.12.1). (15.1.6) We have already seen (11.1) that the algebra J£?(E) of continuous endomorphisms of a complex normed space E is a normed algebra (in general noncommutative) with the identity mapping 1E of E as unit element (the inequality (15.1.1) in this case is just (5.7.5), and the relation (15.1.2) is obvious). Furthermore, if E is a Banach space, then JSf(E) is a Banach algebra (5.7.3). The set of compact operators in J?(E) is a closed two-sided ideal by virtue of (11.2.6) and (11.2.10). (15.1.7) Let G be a separable metrizable locally compact group. The set M£(G) of bounded measures on G, endowed with the convolution product (A, ju) I-H> A * n an<i the norm (13.20.1) is a Banach algebra. For in this case the inequality (15.1.1) is just (14.6.2.1) with n = 2; the unit element se (the Dirac measure at the neutral element e of G) is such that ||ee|| = 1; and Mc(G), being the dual of the normed space *c(G) (13.20.6), is complete (5.7.3). The sub- space of Mc(G) consisting of measures with base a left Haar measure /? on G can be identified (together with its norm) with the Banach space L£(G, /?), because Nx(/) = ||/- ]B|| (13.20.3), and it follows from (14.9.2) that if G is unimodular, L£(G, /?) is a closed two-sided ideal in M£(G). We shall always identify L£(G, /?) with this ideal (the identification of course depends on the choice of/?). Under this identification, Jf C(G) is identified with a subalgebra (14.10.5) of Mc(G), which is not, in general, an ideal (unless G is compact (14.9.1)) and, in general, is not closed, because its closure is Lc(G, /}) (13.11.6)..1.1) \\xy\\ ^ \\x\\ • \\y\\